mnimput: Multivariate Normal Imputation

In mtsdi: Multivariate Time Series Data Imputation

Description

Perform the modified EM algorithm imputation on a normal multivariate dataset

Usage

mnimput(formula, dataset, by = NULL, log = FALSE, log.offset = 1, 
		eps = 1e-3, maxit = 1e2, ts = TRUE, method = "spline", 
		sp.control = list(df = NULL, weights = NULL), ar.control = 
		list(order = NULL, period = NULL), ga.control = list(formula, 
		weights = NULL), f.eps = 1e-6, f.maxit = 1e3, ga.bf.eps = 1e-6, 
		ga.bf.maxit = 1e3, verbose = FALSE, digits = getOption("digits"))

Arguments

formula	formula indicating the missing data frame, for instance, ~X1+X2+X3+...+Xp
dataset	data with missing values to be imputated
by	factor for variance windows. Default is NULL for a single variance matrix
log	logical. If TRUE data will be transformed into log scale. Default is FALSE
log.offset	If log is TRUE, log values will be shifted by this offset. Default is 1
eps	stop criterion
maxit	maximum number of iterations
ts	logical. TRUE if is time series
method	method for univariate time series filtering. It may be smooth, gam or arima. See Details
sp.control	list for Spline smooth control. See Details
ar.control	list for ARIMA fitting control. See Details
ga.control	list for GAM fitting control. See Details
f.eps	convergence criterion for the ARIMA filter. See arima
f.maxit	maximum number of iterations for the ARIMA filter. See arima
ga.bf.eps	covergence criterion for the backfitting algorithm of GAM models. See gam
ga.bf.maxit	maximum number of iterations for the backfitting algorithm of GAM models. See gam
verbose	if TRUE convergence information on each iteration is printed. Default is FALSE
digits	an integer indicating the decimal places. If not supplied, it is taken from options

Details

This is a modified version of the EM algorithm for imputation of missing values. It is also applicable to time series data. When it is explicited the time series attribute through the argument ts, missing values are estimated accounting for both correlation between time series and time structure of the series itself. Several filters can be used for prediction of the mean vector in the E-step.

One can select the method for the univariate time series filtering by the argument method. The default method is "spline". In this case a smooth spline is fitted to each of the time series at each iteration. Some parameters can be passed to smooth.spline through sm.control. df is a vector as long as the number of columns in dataset holding fixed degrees of freedom of the splines. If NULL, the degrees of freedom of each spline are chosen by cross-validation. If df has length 1, this values is recycled for all the covariates. weights must be a matrix of the same size of dataset with the weights for smooth.spline. If NULL, all the observations will have weights equal to 1.

Other possibity for time series filtering is to fitting an ARIMA model for each of the time series by setting method to "arima". The ARIMA models must be identified before using this function, nonetheless. arima function can be partially controlled through ar.control. Each column of order must hold the corresponding (p,d,q) parameters for each univariate time series if period is NULL. If period is not NULL, order must also hold the multiplicative seasonality parameters, so each column of order takes the form (p,d,q,P,D,Q). period is the multiplicative seasonality period. f.eps and f.maxit control de convergence of the ARIMA fitting algorithm. Convergence problems due non stationarity may arise when using this option.

Last but not least, a very interesting approach to modelling temporal patterns to use a full fledged regression model. It is possible to use generalised aditive (or linear) models with exogenous variates to proper filtering of time patterns. One must set method to gam and supply a vector of formulas in ga.control. One must supply one formula for each covariate. Using covariates that are part of the formula of the imputation model may yield some colinearity among the variates. See gam and glm for details. In order to use regression models for the level, set method to "gam"

Simulations have shown that the algorithm is stable and yields good results on imputation of normal data.

Value

The function returns an object of class mtsdi containing

call	function call
dataset	imputed dataset
muhat	estimated mean vector
sigmahat	estimated covariance matrix
missings	vector holding the number of missing values on each row
iterations	number of iterations until convergence or reach maxit
convergence	convergence value. See Details
converged	a logical indicating if the algorithm converged
time	elapsed time of the process

Author(s)

Washington Junger [e-mail] and Antonio Ponce de Leon [e-mail]

References

Junger, W.L. and Ponce de Leon, A. (2015) Imputation of Missing Data in Time Series for Air Pollutants. Atmospheric Environment, 102, 96-104.

Johnson, R., Wichern, D. (1998) Applied Multivariate Statistical Analysis. Prentice Hall.

Dempster, A., Laird, N., Rubin, D. (1977) Maximum Likelihood from Incomplete Data via the Algorithm EM. Journal of the Royal Statistical Society 39(B)), 1–38.

McLachlan, G. J., Krishnan, T. (1997) The EM algorithm and extensions. John Wiley and Sons.

Box, G., Jenkins, G., Reinsel, G. (1994) Time Series Analysis: Forecasting and Control. 3 ed. Prentice Hall.

Hastie, T. J.; Tibshirani, R. J. (1990) Generalized Additive Models. Chapman and Hall.

See Also

mnimput, predict.mtsdi, edaprep

Examples

data(miss)
f <- ~c31+c32+c33+c34+c35
## one-window covariance
i <- mnimput(f,miss,eps=1e-3,ts=TRUE, method="spline",sp.control=list(df=c(7,7,7,7,7)))
summary(i)

## two-window covariances
b<-c(rep("year1",12),rep("year2",12))
ii <- mnimput(f,miss,by=b,eps=1e-3,ts=TRUE, method="spline",sp.control=list(df=c(7,7,7,7,7)))
summary(ii)

Example output

Loading required package: gam
Loading required package: splines
Loading required package: foreach
Loaded gam 1.16

mtsdi 0.3.5

Call:
mnimput(formula = f, dataset = miss, eps = 0.001, ts = TRUE, 
    method = "spline", sp.control = list(df = c(7, 7, 7, 7, 7)))

Estimated mean vector:
     c31      c32      c33      c34      c35 
4.428260 3.404179 3.604979 4.111231 4.231927 

Estimated covariance matrix:
          c31       c32       c33      c34      c35
c31 13.198835  9.821802  9.880967 12.48171 15.00930
c32  9.821802 10.565439 13.290885 12.54823 13.24290
c33  9.880967 13.290885 23.171432 15.31136 18.71416
c34 12.481710 12.548226 15.311356 16.06571 16.86641
c35 15.009299 13.242895 18.714161 16.86641 23.27084

Data are on the original scale.

The algorithm converged after 9 iterations with relative diference in covariance matrix equal to 0.0007446.
The process took 00:00:00.

Time filtering models:

Filter model for variate: c31
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 146.3098
GCV: 12.15173


Filter model for variate: c32
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 115.0795
GCV: 9.557904


Filter model for variate: c33
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 214.1316
GCV: 17.78465


Filter model for variate: c34
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 166.8908
GCV: 13.86108


Filter model for variate: c35

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 278.771
GCV: 23.15326

Call:
mnimput(formula = f, dataset = miss, by = b, eps = 0.001, ts = TRUE, 
    method = "spline", sp.control = list(df = c(7, 7, 7, 7, 7)))

Estimated mean vector:

Covariance window id: year1
     c31      c32      c33      c34      c35 
4.439354 3.058333 2.943321 4.332500 4.282500 

Covariance window id: year2
     c31      c32      c33      c34      c35 
4.413333 3.772190 3.107810 3.900743 4.054827 

Estimated covariance matrix:

BY factor level: year1
         c31      c32      c33      c34      c35
c31 17.06912 12.75444 14.12697 17.35886 22.27431
c32 12.75444 11.35351 11.27189 14.98227 15.55464
c33 14.12697 11.27189 12.79017 15.12799 18.09330
c34 17.35886 14.98227 15.12799 20.27335 21.51136
c35 22.27431 15.55464 18.09330 21.51136 31.15114

BY factor level: year2
          c31       c32        c33       c34       c35
c31 9.1459222  6.956757  0.8428413  7.089435  6.179044
c32 6.9567566  9.612193 14.3186661 10.589023 11.390653
c33 0.8428413 14.318666 46.1144209 16.362282 22.125969
c34 7.0894347 10.589023 16.3622818 12.706236 13.246699
c35 6.1790439 11.390653 22.1259691 13.246699 15.542236

Data are on the original scale.

The algorithm converged after 53 iterations with relative diference in covariance matrix equal to 0.0008958.
The process took 00:00:00.

Time filtering models:

Filter model for variate: c31
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 146.2754
GCV: 12.14887


Filter model for variate: c32
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 115.6142
GCV: 9.602314


Filter model for variate: c33
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 238.2249
GCV: 19.78572


Filter model for variate: c34
Call:
smooth.spline(x = t, y = xn[, j], w = w[, j], df = df[j])

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 161.1101
GCV: 13.38096


Filter model for variate: c35

Smoothing Parameter  spar= 0.5179712  lambda= 0.0003185393 (12 iterations)
Equivalent Degrees of Freedom (Df): 7.001002
Penalized Criterion (weighted RSS): 247.7141
GCV: 20.57384

mtsdi documentation built on May 2, 2019, 1:09 p.m.